| [section:geometric_dist Geometric Distribution] |
| |
| ``#include <boost/math/distributions/geometric.hpp>`` |
| |
| namespace boost{ namespace math{ |
| |
| template <class RealType = double, |
| class ``__Policy`` = ``__policy_class`` > |
| class geometric_distribution; |
| |
| typedef geometric_distribution<> geometric; |
| |
| template <class RealType, class ``__Policy``> |
| class geometric_distribution |
| { |
| public: |
| typedef RealType value_type; |
| typedef Policy policy_type; |
| // Constructor from success_fraction: |
| geometric_distribution(RealType p); |
| |
| // Parameter accessors: |
| RealType success_fraction() const; |
| RealType successes() const; |
| |
| // Bounds on success fraction: |
| static RealType find_lower_bound_on_p( |
| RealType trials, |
| RealType successes, |
| RealType probability); // alpha |
| static RealType find_upper_bound_on_p( |
| RealType trials, |
| RealType successes, |
| RealType probability); // alpha |
| |
| // Estimate min/max number of trials: |
| static RealType find_minimum_number_of_trials( |
| RealType k, // Number of failures. |
| RealType p, // Success fraction. |
| RealType probability); // Probability threshold alpha. |
| static RealType find_maximum_number_of_trials( |
| RealType k, // Number of failures. |
| RealType p, // Success fraction. |
| RealType probability); // Probability threshold alpha. |
| }; |
| |
| }} // namespaces |
| |
| The class type `geometric_distribution` represents a |
| [@http://en.wikipedia.org/wiki/geometric_distribution geometric distribution]: |
| it is used when there are exactly two mutually exclusive outcomes of a |
| [@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trial]: |
| these outcomes are labelled "success" and "failure". |
| |
| For [@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials] |
| each with success fraction /p/, the geometric distribution gives |
| the probability of observing /k/ trials (failures, events, occurrences, or arrivals) |
| before the first success. |
| |
| [note For this implementation, the set of trials *includes zero* |
| (unlike another definition where the set of trials starts at one, sometimes named /shifted/).] |
| The geometric distribution assumes that success_fraction /p/ is fixed for all /k/ trials. |
| |
| The probability that there are /k/ failures before the first success is |
| |
| __spaces Pr(Y=/k/) = (1-/p/)[super /k/]/p/ |
| |
| For example, when throwing a 6-face dice the success probability /p/ = 1/6 = 0.1666[recur][space]. |
| Throwing repeatedly until a /three/ appears, |
| the probability distribution of the number of times /not-a-three/ is thrown |
| is geometric. |
| |
| Geometric distribution has the Probability Density Function PDF: |
| |
| __spaces (1-/p/)[super /k/]/p/ |
| |
| The following graph illustrates how the PDF and CDF vary for three examples |
| of the success fraction /p/, |
| (when considering the geometric distribution as a continuous function), |
| |
| [graph geometric_pdf_2] |
| |
| [graph geometric_cdf_2] |
| |
| and as discrete. |
| |
| [graph geometric_pdf_discrete] |
| |
| [graph geometric_cdf_discrete] |
| |
| |
| [h4 Related Distributions] |
| |
| The geometric distribution is a special case of |
| the __negative_binomial_distrib with successes parameter /r/ = 1, |
| so only one first and only success is required : thus by definition |
| __spaces `geometric(p) == negative_binomial(1, p)` |
| |
| negative_binomial_distribution(RealType r, RealType success_fraction); |
| negative_binomial nb(1, success_fraction); |
| geometric g(success_fraction); |
| ASSERT(pdf(nb, 1) == pdf(g, 1)); |
| |
| This implementation uses real numbers for the computation throughout |
| (because it uses the *real-valued* power and exponential functions). |
| So to obtain a conventional strictly-discrete geometric distribution |
| you must ensure that an integer value is provided for the number of trials |
| (random variable) /k/, |
| and take integer values (floor or ceil functions) from functions that return |
| a number of successes. |
| |
| [discrete_quantile_warning geometric] |
| |
| [h4 Member Functions] |
| |
| [h5 Constructor] |
| |
| geometric_distribution(RealType p); |
| |
| Constructor: /p/ or success_fraction is the probability of success of a single trial. |
| |
| Requires: `0 <= p <= 1`. |
| |
| [h5 Accessors] |
| |
| RealType success_fraction() const; // successes / trials (0 <= p <= 1) |
| |
| Returns the success_fraction parameter /p/ from which this distribution was constructed. |
| |
| RealType successes() const; // required successes always one, |
| // included for compatibility with negative binomial distribution |
| // with successes r == 1. |
| |
| Returns unity. |
| |
| The following functions are equivalent to those provided for the negative binomial, |
| with successes = 1, but are provided here for completeness. |
| |
| The best method of calculation for the following functions is disputed: |
| see __binomial_distrib and __negative_binomial_distrib for more discussion. |
| |
| [h5 Lower Bound on success_fraction Parameter ['p]] |
| |
| static RealType find_lower_bound_on_p( |
| RealType failures, |
| RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. |
| |
| Returns a *lower bound* on the success fraction: |
| |
| [variablelist |
| [[failures][The total number of failures before the 1st success.]] |
| [[alpha][The largest acceptable probability that the true value of |
| the success fraction is [*less than] the value returned.]] |
| ] |
| |
| For example, if you observe /k/ failures from /n/ trials |
| the best estimate for the success fraction is simply 1/['n], but if you |
| want to be 95% sure that the true value is [*greater than] some value, |
| ['p[sub min]], then: |
| |
| p``[sub min]`` = geometric_distribution<RealType>:: |
| find_lower_bound_on_p(failures, 0.05); |
| |
| [link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.] |
| |
| This function uses the Clopper-Pearson method of computing the lower bound on the |
| success fraction, whilst many texts refer to this method as giving an "exact" |
| result in practice it produces an interval that guarantees ['at least] the |
| coverage required, and may produce pessimistic estimates for some combinations |
| of /failures/ and /successes/. See: |
| |
| [@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf |
| Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions. |
| Computational statistics and data analysis, 2005, vol. 48, no3, 605-621]. |
| |
| [h5 Upper Bound on success_fraction Parameter p] |
| |
| static RealType find_upper_bound_on_p( |
| RealType trials, |
| RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence. |
| |
| Returns an *upper bound* on the success fraction: |
| |
| [variablelist |
| [[trials][The total number of trials conducted.]] |
| [[alpha][The largest acceptable probability that the true value of |
| the success fraction is [*greater than] the value returned.]] |
| ] |
| |
| For example, if you observe /k/ successes from /n/ trials the |
| best estimate for the success fraction is simply ['k/n], but if you |
| want to be 95% sure that the true value is [*less than] some value, |
| ['p[sub max]], then: |
| |
| p``[sub max]`` = geometric_distribution<RealType>::find_upper_bound_on_p( |
| k, 0.05); |
| |
| [link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.] |
| |
| This function uses the Clopper-Pearson method of computing the lower bound on the |
| success fraction, whilst many texts refer to this method as giving an "exact" |
| result in practice it produces an interval that guarantees ['at least] the |
| coverage required, and may produce pessimistic estimates for some combinations |
| of /failures/ and /successes/. See: |
| |
| [@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf |
| Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions. |
| Computational statistics and data analysis, 2005, vol. 48, no3, 605-621]. |
| |
| [h5 Estimating Number of Trials to Ensure at Least a Certain Number of Failures] |
| |
| static RealType find_minimum_number_of_trials( |
| RealType k, // number of failures. |
| RealType p, // success fraction. |
| RealType alpha); // probability threshold (0.05 equivalent to 95%). |
| |
| This functions estimates the number of trials required to achieve a certain |
| probability that [*more than ['k] failures will be observed]. |
| |
| [variablelist |
| [[k][The target number of failures to be observed.]] |
| [[p][The probability of ['success] for each trial.]] |
| [[alpha][The maximum acceptable ['risk] that only ['k] failures or fewer will be observed.]] |
| ] |
| |
| For example: |
| |
| geometric_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05); |
| |
| Returns the smallest number of trials we must conduct to be 95% (1-0.05) sure |
| of seeing 10 failures that occur with frequency one half. |
| |
| [link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.] |
| |
| This function uses numeric inversion of the geometric distribution |
| to obtain the result: another interpretation of the result is that it finds |
| the number of trials (failures) that will lead to an /alpha/ probability |
| of observing /k/ failures or fewer. |
| |
| [h5 Estimating Number of Trials to Ensure a Maximum Number of Failures or Less] |
| |
| static RealType find_maximum_number_of_trials( |
| RealType k, // number of failures. |
| RealType p, // success fraction. |
| RealType alpha); // probability threshold (0.05 equivalent to 95%). |
| |
| This functions estimates the maximum number of trials we can conduct and achieve |
| a certain probability that [*k failures or fewer will be observed]. |
| |
| [variablelist |
| [[k][The maximum number of failures to be observed.]] |
| [[p][The probability of ['success] for each trial.]] |
| [[alpha][The maximum acceptable ['risk] that more than ['k] failures will be observed.]] |
| ] |
| |
| For example: |
| |
| geometric_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05); |
| |
| Returns the largest number of trials we can conduct and still be 95% sure |
| of seeing no failures that occur with frequency one in one million. |
| |
| This function uses numeric inversion of the geometric distribution |
| to obtain the result: another interpretation of the result, is that it finds |
| the number of trials that will lead to an /alpha/ probability |
| of observing more than k failures. |
| |
| [h4 Non-member Accessors] |
| |
| All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] |
| that are generic to all distributions are supported: __usual_accessors. |
| |
| However it's worth taking a moment to define what these actually mean in |
| the context of this distribution: |
| |
| [table Meaning of the non-member accessors. |
| [[Function][Meaning]] |
| [[__pdf] |
| [The probability of obtaining [*exactly k failures] from /k/ trials |
| with success fraction p. For example: |
| |
| ``pdf(geometric(p), k)``]] |
| [[__cdf] |
| [The probability of obtaining [*k failures or fewer] from /k/ trials |
| with success fraction p and success on the last trial. For example: |
| |
| ``cdf(geometric(p), k)``]] |
| [[__ccdf] |
| [The probability of obtaining [*more than k failures] from /k/ trials |
| with success fraction p and success on the last trial. For example: |
| |
| ``cdf(complement(geometric(p), k))``]] |
| [[__quantile] |
| [The [*greatest] number of failures /k/ expected to be observed from /k/ trials |
| with success fraction /p/, at probability /P/. Note that the value returned |
| is a real-number, and not an integer. Depending on the use case you may |
| want to take either the floor or ceiling of the real result. For example: |
| ``quantile(geometric(p), P)``]] |
| [[__quantile_c] |
| [The [*smallest] number of failures /k/ expected to be observed from /k/ trials |
| with success fraction /p/, at probability /P/. Note that the value returned |
| is a real-number, and not an integer. Depending on the use case you may |
| want to take either the floor or ceiling of the real result. For example: |
| ``quantile(complement(geometric(p), P))``]] |
| ] |
| |
| [h4 Accuracy] |
| |
| This distribution is implemented using the pow and exp functions, so most results |
| are accurate within a few epsilon for the RealType. |
| For extreme values of `double` /p/, for example 0.9999999999, |
| accuracy can fall significantly, for example to 10 decimal digits (from 16). |
| |
| [h4 Implementation] |
| |
| In the following table, /p/ is the probability that any one trial will |
| be successful (the success fraction), /k/ is the number of failures, |
| /p/ is the probability and /q = 1-p/, |
| /x/ is the given probability to estimate |
| the expected number of failures using the quantile. |
| |
| [table |
| [[Function][Implementation Notes]] |
| [[pdf][pdf = p * pow(q, k)]] |
| [[cdf][cdf = 1 - q[super k=1]]] |
| [[cdf complement][exp(log1p(-p) * (k+1))]] |
| [[quantile][k = log1p(-x) / log1p(-p) -1]] |
| [[quantile from the complement][k = log(x) / log1p(-p) -1]] |
| [[mean][(1-p)/p]] |
| [[variance][(1-p)/p[sup2]]] |
| [[mode][0]] |
| [[skewness][(2-p)/[sqrt]q]] |
| [[kurtosis][9+p[sup2]/q]] |
| [[kurtosis excess][6 +p[sup2]/q]] |
| [[parameter estimation member functions][See __negative_binomial_distrib]] |
| [[`find_lower_bound_on_p`][See __negative_binomial_distrib]] |
| [[`find_upper_bound_on_p`][See __negative_binomial_distrib]] |
| [[`find_minimum_number_of_trials`][See __negative_binomial_distrib]] |
| [[`find_maximum_number_of_trials`][See __negative_binomial_distrib]] |
| ] |
| |
| [endsect][/section:geometric_dist geometric] |
| |
| [/ geometric.qbk |
| Copyright 2010 John Maddock and Paul A. Bristow. |
| Distributed under the Boost Software License, Version 1.0. |
| (See accompanying file LICENSE_1_0.txt or copy at |
| http://www.boost.org/LICENSE_1_0.txt). |
| ] |
| |